Ordinary Differential Equations I
Ordinary Differential Equations I MA 532
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This 3 page Class Notes was uploaded by Braeden Lind on Thursday October 15, 2015. The Class Notes belongs to MA 532 at North Carolina State University taught by Stephen Schecter in Fall. Since its upload, it has received 6 views. For similar materials see /class/223710/ma-532-north-carolina-state-university in Mathematics (M) at North Carolina State University.
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Date Created: 10/15/15
Material on the MA 532 Final Exam Steve Soheoter November 25 2003 H One dimensional ODEs phase portraits limits bifurcation diagrams Supplementary Problems 1 problems 1 4 2 5039 fac phase portraits Supplementary Problems 1 problems 6 7 9 Existence uniqueness ow property Supplementary Problems 2 7 Stability Liapunov functions Lasalle7s lnvariance Principle Exercises 135 and 136 Suplementary Problems 3 problems 1 3 Cf Gro39nwall7s inequality linear differential equations Liouville7s formula Exercises 22 23 229 Supplementary Problems 4 problems 173 03 dc Ax Exercise 226 Supplementary Problems 5 and 6 5 Manifolds Exercise 160 Supplementary Problems 7 problem 1 00 Analyzing equilibria by linearization analyzing degenerate equilibria in the plane using polar coordinates Exercise 129 Supplementary Problems 7 problems 274 3 Analyzing planar vector elds 77at in nity77 Supplementary Problems 7 problem 5 Supplementary Problems 8 problem 1 10 Periodic solutions of time periodic differential equations Supplementary Problems 8 problems 278 11 Using the Poincar Bendixson Theorem the linear variational equation and Bendix son7s Criterion to study closed orbits of differential equations in the plane Exercises 1138 rst sentence 1144 Personal help sheet You may bring one sheet of paper with whatever you nd helpful written on it You may use both sides 1 3 9 MA 532 Supplementary Problems 8 corrected December 4 2003 Based on exercise 194 Consider the differential equation H V 2y2717 y 5xy71 A D a Show that there are no equilibria b Use the method of nullclines to draw the phase portrait in the nite plane Draw the curves where at 0 and y 0 Determine the signs of at and y in the regions between these curves Use this information to draw the phase portrait c Show that there is a orbit l for which 21 3 CC d On Par what does approach as no a 700 On l1 what does approach as 90 a 007 Try to answer these questions by combining information from parts b and unique orbit Dr for which a 0 as 90 a 00 and a unique a 0 as 90 a 700 Suggestion use the coordinates u i Suppose that bt is a 27T periodic continuous function and let be 02quot 95 d5 Show that all solutions of 50 bt are 27T periodic if he 0 otherwise7 they are all un bounded Hint Show that the Poincare map is 135 5 be Fredholm s Alternative Suppose that at and bt are 27T periodic continuous func tions7 and let a0 02quot 15 d5 Show the following properties of the differential equa tion at atx bt a If 10 3A 07 then there is a unique 27T periodic orbit It is asymptotically stable if 10 lt 07 and asymptotically unstable if 10 gt 0 b Let Go 02quot exp s27r au du b5 d5 If 10 07 then every solution is 27T periodic if and only if 00 0 c If 10 07 then every solution is unbounded if 00 3A 0 Hint Show using the variation of constants formula that the Poincare map is 27r 27r 135 6105 exp aw dune ds 0 s and 135 5 if and only if 1 7 e 0 Co 1 7 Cf 03 5 00 Show that the differential equation at 7x5 Ct7 where Ct is a 27T periodic continu ous function7 has a 27T periodic solution Show that any such solution is asymptotically stable Explain why this implies that there is only one 27T periodic solution Let Ct7 dt7 and 6t be 27T periodic continuous functions Show that the differential equation 90 7x3 Ctx2 dtx 6t has at least one 27T periodic solution Show that if this solution is unstable7 then there must be another 27T periodic solution Hint Show that at is negative if 90 is suf ciently positive7 and positive if 90 is suf ciently negative Riccatz39 Equation Suppose that at and bt are 27T periodic continuous functions Prove that the Riccati equation at bt atx 7 x2 has at most two 27T periodic solutions Hint Suppose that gtt is a 27T periodic solu tion lf 90t is another solution7 let yt 90t 7 gtt Show that y CW 7 127 Show that where Ct at 7 2 gtt Then let wt w 70tw 1 Use the Fredholrn Alternative to discuss separately the cases 02quot Ct dt 3A 0 and 02quot Ct dt 0 Show that 90 sint is a 27T periodic solution of the differential equation 50 7x3 2x sin3t 7 2sint cost Show that this solution is unstable with P 0 6 How many more 27T periodic solutions can you guarantee Suppose that at is 27T periodic with 0 lt at lt 1 for all t Show that the differential equation at 9090 7 atx 7 1 has at least three 27T periodic solutions Hint Show that 90t 0 and 90t 1 are asyrnptotically stable7 and explain why this implies that there is a 27T periodic solution between them
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